Kloosterman sums in residue rings

TL;DR

This paper extends results on Kloosterman sums from prime moduli to general moduli, establishing additive properties of reciprocal sets and applying bounds to improve estimates in number theory.

Contribution

It introduces new bounds on multilinear exponential sums for general moduli and applies them to classical problems like Brun-Titchmarsh and short Kloosterman sums.

Findings

Generalized Kloosterman sum bounds to composite moduli

Established additive properties of reciprocal sets in residue rings

Improved estimates for short Kloosterman sums

Abstract

In the present paper, we generalize some of the results on Kloosterman sums proven in \cite{BG} for prime moduli to general moduli. This requires to establish the corresponding additive properties of the reciprocal set $$ I^{-1}=\{x^{-1}:\quad x\in I\}, $$ where $I$ is an interval in the ring of residue classes modulo a large positive integer. We apply our bounds on multilinear exponential sums to the Brun-Titchmarsh theorem and the estimate of very short Kloosterman sums, hence generalizing our earlier work to the setting of general modulus.